Consider $X$ a compact manifold. Let $S(-)$ denotes suspension of $(-)$. Then $Vect_n(S(X))\cong [X,GL(n,C)]$ where $[-,-]$ denotes homotopy maps. Clearly there will be $\otimes,\oplus$ defined on graded ring $\oplus_i Vect_i(S(X))$. Since $GL(n,C)\subset\lim_iGL(i,C)$ where $lim_i$ is over direct limit over inclusion, I would expect $[X,GL(n,C)]\to [X,\lim_i GL(i,C)]$ which is inclusion. Since $X$ is compact, the image of $X$ will eventually falls into a finite index $i$. Thus $\lim_i[X,GL(i,C)]\to [X,\lim_i GL(i,C)]$ is epimorphism.
$\textbf{Q1:}$ Is $\lim_i[X,GL(i,C)]\to [X,\lim_i GL(i,C)]$ mono? There is no particular reason that this is mono as it seems that domain set is much larger and direct limit can't be pulled out from second slot of $Hom(-,\lim_i-)$.
$\textbf{Q2:}$ Due to surjectivity of previous map, I can push multiplication and addition structure to $[X,\lim_iGL(i,C)]$ with addition as block diagonal matrices and multiplication as kronecker product. Is this the ring structure on $[X,\lim_i GL(i,C)]$?
Since $Gl_n(\mathbb{C})$ deformation retracts on $U_n$, you might as well replace it with this space. This isn't necessesary in the following, but I find it more convenient to work with compact CW complexes.
The sequence
$$\dots\rightarrow U_n\rightarrow U_{n+1}\rightarrow\dots$$
is now a sequence of closed cofibrations ($U_n\subseteq U_{n+1}$ is a closed Lie subgroup), so in particular you can replace the direct limit aka colimit with the telescope
$$tel(U_n)=\bigcup U_n\times[n,n+1]\bigg/\sim.$$
More specifically I mean that the canonical map $tel(U_n)\rightarrow colim\;U_n$ is a homotopy equivalence. The details are recounted in Hatcher's book somewhere if I recall.
Now, there is a canonical comparisson map
$$colim\;[X,U_n]\rightarrow [X,tel(U_n)]$$
and its not difficult to see that it is bijecive when $X$ is a compact CW complex.In particular when $X$ is compact manifold. The idea is simple, any map $X\rightarrow tel(U_n)$ factors through some finite stage of the telescope by compactness, as does any homotopy of said map $X\times I\rightarrow tel(U_n)$.
As for question $2$, I don't see why the ring structure you describe would necessarily be unique, but it certainly seems to me that you are indeed describing the standard ring structure on $\widetilde K^1X$, which is exactly the object in consideration here.
To see what your ring structure is explicitly it's best to consider the universal case. That is, to define addition and multiplication maps $$U\times U\xrightarrow{\oplus} U\qquad U\times U\xrightarrow{\otimes} U$$ on $U=colim\;U_n$. These then give the ring structure on $[X,U]$ in the standard way.
The point is that your diagram is sufficently cofibrant so as to be able to just take hocolims of the block addition and tensor product maps $$U_m\times U_n\xrightarrow{\oplus} U_{n+m},\qquad U_m\times U_n\xrightarrow\otimes U_{nm}$$ to get your addition and multiplication maps on $U$.
There are just two nontrivial points. Firstly you need to have been clever enough with your definitions to make sure that the block addition and tensor product maps are suffiently coherently for this to make sense. I like the book 'Topology of Lie Groups, II' by Mimura and Toda for the details here. The second point is to exactly why the direct homotopy limits should be compatible with the products in the domains of the above maps, but this is a standard result.